AI 中文总结
该研究提出改进Bryson-Frazier平滑器替代RTS平滑器，可避免协方差矩阵求逆的数值问题，还能复用中间量实现低额外成本的核超参数学习，为一维高斯过程回归提供了统一的稳健推理方法。
AI 中文摘要
具有平稳可积分核函数的一维高斯过程可接受精确或任意精度的状态空间表示，能通过卡尔曼滤波和Rauch-Tung-Striebel（RTS）平滑器实现线性时间推理。但RTS平滑器需要对预测状态协方差矩阵求逆，该操作可能出现病态情况，进而导致数值不稳定性。本研究重新探讨改进Bryson-Frazier（MBF）平滑器，将其作为RTS平滑器的替代方案，用于状态空间表示下的高斯过程回归。MBF平滑器除了能降低计算成本和内存需求外，还能计算出与RTS平滑器相同的后验分布，同时避免了有问题的协方差矩阵求逆及相关的数值不稳定性。此外，研究表明MBF平滑器计算的中间量可重复使用，用于计算负对数边缘似然的梯度，从而以极少的额外成本实现核超参数学习。这些结果共同确立了MBF平滑器作为一维高斯过程回归的推理与核超参数学习的统一且数值稳健的方法。
英文摘要
One-dimensional Gaussian processes with stationary, integrable kernel functions admit exact or arbitrarily accurate state-space representations, enabling linear-time inference through Kalman filtering and Rauch-Tung-Striebel (RTS) smoothing. However, the RTS smoother requires inversion of predicted state covariance matrices, which can become ill-conditioned and may therefore lead to numerical instabilities. In this work, we revisit the modified Bryson-Frazier (MBF) smoother as an alternative to the RTS smoother for Gaussian process regression in its state-space representation. In addition to reducing computational cost and memory requirements, the MBF smoother computes the same posterior distributions as the RTS smoother while avoiding the problematic covariance matrix inversion and the associated numerical instabilities. Furthermore, we demonstrate that the intermediate quantities computed by the MBF smoother can be reused to compute gradients of the negative log marginal likelihood, enabling kernel hyperparameter learning with minimal additional cost. Together, these results establish the MBF smoother as a unified and numerically robust approach to inference and kernel hyperparameter learning for one-dimensional Gaussian process regression.
CommentsAccepted at the Second International Conference on Probabilistic Numerics (ProbNum 2026)
